Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10)(11,12)(13,15)(14,21)(16,18)(17,22)(19,24)(20,23)(25,26)(27,34)(28,32)(29,31)(30,35)(33,38)(36,40)(37,39);(1,2,5,3)(4,6,8,7)(9,11,15,16)(10,12,13,18)(14,19,22,23)(17,20,21,24)(25,27,30,29)(26,31,35,34)(28,36,38,37)(32,39,33,40);(1,2,5,3)(4,6,8,7)(9,12,15,18)(10,16,13,11)(14,20,22,24)(17,23,21,19)(25,28,30,38)(26,32,35,33)(27,36,29,37)(31,39,34,40);(2,3)(6,7)(9,12)(10,11)(13,16)(14,20)(15,18)(17,19)(21,23)(22,24)(25,28)(26,33)(27,36)(29,37)(30,38)(31,40)(32,35)(34,39);(1,3,5,2)(4,7,8,6)(9,13,15,10)(11,18,16,12)(14,21,22,17)(19,24,23,20)(25,26)(27,31)(28,32)(29,34)(30,35)(33,38)(36,39)(37,40);(1,4)(2,6)(3,7)(5,8)(9,14,15,22)(10,17,13,21)(11,19,16,23)(12,20,18,24)(25,29,30,27)(26,34,35,31)(28,37,38,36)(32,40,33,39)Polycyclic presentation:
2^9|b^2=gi,c^2=gi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8066215);